Laminar burning velocity

For a one-step irreversible chemistry, i.e., ${\displaystyle \nu _{F}{\rm {{F}+\nu _{O}{\rm {{O}_{2}\rightarrow {\rm {Products}}}}}}}$, the planar, adiabatic flame has explicit expression for the burning velocity derived from activation energy asymptotics when the Zel'dovich number ${\displaystyle \beta \gg 1.}$ The reaction rate ${\displaystyle \omega }$ (number of moles of fuel consumed per unit volume per unit time) is taken to be Arrhenius form,

${\displaystyle \omega =B\left({\frac {\rho Y_{F}}{W_{F}}}\right)^{m}\left({\frac {\rho Y_{O_{2}}}{W_{O_{2}}}}\right)^{n}e^{-E_{a}/RT},}$

where ${\displaystyle B}$ is the pre-exponential factor, ${\displaystyle \rho }$ is the density, ${\displaystyle Y_{F}}$ is the fuel mass fraction, ${\displaystyle Y_{O_{2}}}$ is the oxidizer mass fraction, ${\displaystyle E_{a}}$ is the activation energy, ${\displaystyle R}$ is the universal gas constant, ${\displaystyle T}$ is the temperature, ${\displaystyle W_{F}\ \&\ W_{O_{2}}}$ are the molecular weights of fuel and oxidizer, respectively and ${\displaystyle m\ \&\ n}$ are the reaction orders. Let the unburnt conditions far ahead of the flame be denoted with subscript ${\displaystyle u}$ and similarly, the burnt gas conditions by ${\displaystyle b}$, then we can define an equivalence ratio ${\displaystyle \phi }$ for the unburnt mixture as

${\displaystyle \phi ={\frac {\nu _{O_{2}}W_{O_{2}}}{\nu _{F}W_{F}}}{\frac {Y_{F,u}}{Y_{O_{2},u}}}}$.

Then the planar laminar burning velocity for fuel-rich mixture (${\displaystyle \phi >1}$) is given by^[2][3]

${\displaystyle S_{L}=\left\{{\frac {2B\lambda _{b}\rho _{b}^{m+n}\nu _{F}^{m}Y_{O_{2},u}^{m+n-1}G(n,m,a)}{c_{p,b}\rho _{u}^{2}\nu _{O_{2}}W_{O_{2}}^{m+n-1}\beta ^{m+n+1}\mathrm {Le} _{O_{2}}^{-n}\mathrm {Le} _{F}^{-m}}}\right\}^{1/2}e^{-E_{a}/2RT_{b}}+O(\beta ^{-1}),}$

where

${\displaystyle G(n,m,a)=\int _{0}^{\infty }y^{n}(y+a)^{m}\ dy}$

and ${\displaystyle a=\beta (\phi -1)/\mathrm {Le} _{F}}$. Here ${\displaystyle \lambda }$ is the thermal conductivity, ${\displaystyle c_{p}}$ is the specific heat at constant pressure and ${\displaystyle \mathrm {Le} }$ is the Lewis number. Similarly one can write the formula for lean ${\displaystyle \phi <1}$ mixtures. This result is first obtained by T. Mitani in 1980.^[4] Second order correction to this formula with more complicated transport properties were derived by Forman A. Williams and co-workers in the 80s.^[5][6][7]

Variations in local propagation speed of a laminar flame arise due to what is called flame stretch. Flame stretch can happen due to the straining by outer flow velocity field or the curvature of flame; the difference in the propagation speed from the corresponding laminar speed is a function of these effects and may be written as: ^[8][9]

${\displaystyle S_{T}=S_{L}+{\mathcal {M}}_{c}\delta _{L}(S_{L}-\mathbf {v} \cdot \mathbf {n} )\nabla \cdot \mathbf {n} -{\mathcal {M}}_{t}\delta _{L}\nabla _{t}\cdot \mathbf {v} _{t}}$

where ${\displaystyle \mathbf {n} }$ is the unit normal to the flame surface (pointing towards the burnt gas side), ${\displaystyle \mathbf {v} }$ is the flow velocity field evaluated at the flame surface and ${\displaystyle \nabla _{t}\cdot \mathbf {v} _{t}}$ is the surface divergence of the tangential velocity ${\displaystyle \mathbf {v} _{t}=(\mathbf {I} -\mathbf {n} \otimes \mathbf {n} )\mathbf {v} }$; with flow being incompressible outside the flame, ${\displaystyle \nabla _{t}\cdot \mathbf {v} _{t}=-\mathbf {n} \otimes \mathbf {n}$ :\nabla \mathbf {v} -(\mathbf {v} \cdot \mathbf {n} )\nabla \cdot \mathbf {n} } . Moreover, ${\displaystyle {\mathcal {M}}_{c}}$ and ${\displaystyle {\mathcal {M}}_{t}}$ are the two Markstein numbers, associated with the curvature and tangential straining.^[10]